2 Answers
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Neither Voyager 1 nor Voyager 2 was aimed at any particular target outside the Solar System. Their trajectories were largely determined by the requirement to do fly-bys of Jupiter and Saturn (and, in the case of Voyager 2's extended mission, Uranus and Neptune). They've been sending back some interesting results about the boundary between the Solar System and interstellar space, but are not expected to remain operational long enough to send back any data about any other stars.

Voyager 1 happens to be heading "in the general direction of the Solar Apex (the direction of the Sun's motion relative to nearby stars)", which means it should reach the heliopause somewhat sooner than if it were going in a different direction. I don't think that was deliberate. To do that intentionally, they would have had to alter its course as it passed through the Saturn system; as far as I know, its course was optimized for observations of Saturn, its rings, and its moons. And Voyager 2 passed through the termination shock 10 AUs closer to the Sun than Voyager 1 did.

I don't think there's any particular reason to think that either one of them will encounter the black hole at the core of the galaxy. To reach the galactic core, they'd have to have enough velocity to cancel the Sun's orbital motion around the core, about 251 kilometers per second, compared to their actual Sun-relative speed of about 17 kilometers per second.

Both Voyagers are headed towards the outer boundary of the solar
system in search of the heliopause, the region where the Sun's
influence wanes and the beginning of interstellar space can be sensed.
The heliopause has never been reached by any spacecraft; the Voyagers
may be the first to pass through this region, which is thought to
exist somewhere from 8 to 14 billion miles from the Sun. This is where
the million-mile-per-hour solar winds slows to about 250,000 miles per
hour—the first indication that the wind is nearing the heliopause. The
Voyagers should cross the heliopause 10 to 20 years after reaching the
termination shock. The Voyagers have enough electrical power and
thruster fuel to operate at least until 2020. By that time, Voyager 1
will be 12.4 billion miles (19.9 billion KM) from the Sun and Voyager
2 will be 10.5 billion miles (16.9 billion KM) away. Eventually, the
Voyagers will pass other stars. In about 40,000 years, Voyager 1 will
drift within 1.6 light years (9.3 trillion miles) of AC+79 3888, a
star in the constellation of Camelopardalis. In some 296,000 years,
Voyager 2 will pass 4.3 light years (25 trillion miles) from Sirius,
the brightest star in the sky . The Voyagers are destined—perhaps
eternally—to wander the Milky Way.

Both Voyagers are, of course, still influenced by the Sun's gravity. Ignoring other forces (which is a good enough approximation for thousands of years), they'll continue to decelerate, asymptotically approaching a speed that depends on the Sun's gravity and their kinetic energy. According to a footnote in this article,

Nice answer - thanks. Do you know what the eventual relative speed of the Voyagers will be (have they already essentially escaped the sun at 17 km/s?) and how it compares to the other random motions of nearby stars? Will they race out of the local area, or more-or-less-randomly mingle with the others?
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nealmcbJan 16 '12 at 5:33

@nealmcb: They've certainly exceeded Solar escape velocity, but of course nothing ever entirely escapes the Sun's gravity. They haven't reached Galactic escape velocity. Ignoring factors other than their motion and the Sun's gravity, they'll asymptotically approach some speed slower than their current speed. I don't know what that speed is; I'll follow up when/if I manage to look it up or figure it out.
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Keith ThompsonJan 16 '12 at 5:38

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@nealmcb: I've updated my answer with information about their asymptotic velocities. Basically, they've already slowed down nearly as much as they ever will.
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Keith ThompsonJan 16 '12 at 8:00

So basically voyager 2 is not really moving at all. Our solar system is traveling away from the center of the galaxy at approximately 35,000 miles per hour and voyager is traveling toward the center of the galaxy or away from the our sun at the same speed.